The low carbon footprint, biodegradability, interesting mechanical properties, and relatively low price are considered some of the reasons for the increased interest in polylactic acid-based (PLA-based) filaments supplied with natural fillers. However, it is essential to recognize that incorporating natural fillers into virgin PLA significantly impacts the printability of the resulting blends. The complex inter-relationship between process, structure, and properties in the context of fused deposition modeling (FDM)-manufactured biocomposites is still not fully understood, which thus often results in decreased reliability of this technology in the context of biocomposites, decreased accuracy, and the increased presence of defects in the manufactured biocomposite samples. In light of these considerations, this study aims to identify the optimal processing parameters for the FDM manufacturing process involving wood-filled PLA biocomposites. This study presents an optimization approach consisting of Grey Relational Analysis in conjunction with the Taguchi orthogonal array. The optimization process has identified the combination of a scanning speed of 70 mm/s, a layer height of 0.1 mm, and a printing temperature of 220 °C as the most optimal, resulting in the highly satisfactory combination of good dimensional accuracy (Dx = 20.115 mm, Dy = 20.556 mm, and Dz = 20.220 mm) and low presence of voids (1.673%). The experimentally determined Grey Relational Grade of the specimen manufactured with the optimized set of process parameters (0.782) was in good agreement with the predicted value (0. 754), substantiating the validity of the optimization process. Additionally, the research compared the efficacy of optimization between the integrated multiparametric method and the conventional monoparametric strategy. The multiparametric method, which combines Grey Relational Analysis with the Taguchi orthogonal array, exhibited superior performance. Although the monoparametric optimization strategy yielded specimens with favorable values for the targeted properties, the analysis of the remaining characteristics uncovered unsatisfactory results. This highlights the potential drawbacks of relying on a singular optimization approach.